Circulant graphs and GCD and LCM of subsets

Given two sets A and B of integers, we consider the problem of finding a set S⊆A of the smallest possible cardinality such the greatest common divisor of the elements of S∪B equals that of those of A∪B. The particular cases of B=∅ and #B=1 are of special interest and have some links with graph theory. We also consider the corresponding question for the least common multiple of the elements. We establish NP-completeness and approximation results for these problems by relating them to the Set Cover Problem.